The Maxwell Equations, the Lorentz Field and the Electromagnetic Nanofield with Regard to the Question of Relativity

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1 The Maxwell Equaions, he Lorenz Field and he Elecromagneic Nanofield wih Regard o he Quesion of Relaiviy Daniele Sasso * Absrac We discuss he Elecromagneic Theory in some main respecs and specifically wih relaion o he quesion of Relaiviy. Le us consider he Maxwell equaions for he represenaion of he elecromagneic field and he Lorenz force for he descripion of he moion of a paricle in a field of magneic inducion. The Lorenz force is also useful for describing he behavior of he elecromagneic field in presence of cu flux, ha is he physical siuaion which happens wih respec o ineracive moving reference sysems. We examine a las physics of he elecromagneic nanofield which is essenial for he definiion of he behaviour of single energy quana ha compose he energy radiaion. Le us have o accep ha physics and science have an insurmounable limi and ha hey can give an answer o all possible quesions excep he firs and he las. Those who wan o give an answer o hese wo quesions have o leave physics and science wih heir mehod and o go ino he he order of philosophy or of religion. 1. Inroducion Elecromagneism is he discipline ha sudies all he elecric and magneic phenomena and i go he highes poin of synhesis in Maxwell s equaions. Those equaions neverheless disregard he Lorenz force ha really is indispensable for describing a few elecromagneic evens in which he law of elecromagneic inducion due o cu flux inervenes. This consideraion can be relaed for example o he moion wheher of a single paricle or of a paricle beam in a field of magneic inducion, bu i is also useful in order o describe he relaivisic behavior of Maxwell s equaions wih respec o Einseinian inerial reference sysems ha are open and herefore ineracive. Elecromagneism is characerized by he propagaion of boh he elecromagneic field and elecromagneic waves. The ligh and energy radiaions wih greaer frequencies han he ligh ( X rays, rays, rays ) aren really elecromagneic waves bu raher beams composed of energy quana. From he physical viewpoin hen he single quanum is an elecromagneic nanowave generaed by an elecromagneic nanofield ha can be described by a group of mahemaical relaions derived from Maxwell s equaions. * e_mail: dgsasso@alice.i This paper was sen o he Physics Colloquium in Porland, Maine, 01 Augus 1

2 . The Maxwell equaions The hisorical group of Maxwell s equaions is defined in a vacuum by he following four equaions, derived from he scienific works by J. C. Maxwell, A Dynamical Theory of he Elecromagneic Field (1865), A Treaise on Elecriciy and Magneism (1873), and from he scienific work by M. Faraday On he Physical Naure of Force Lines (185): div E = (1) div B = 0 () ro E = - B (3) ro B = J + 1 E (4) c Almos all elecric and magneic phenomena can be described by he previous equaions and i is surely a big advanage o have a synhesis of elecromagneism hrough a limied group of equaions. This group of equaions neverheless raised soon a problem because i seemed i didn respec he Principle of Relaiviy. In fac in he equaion (4) he speed of ligh appears: in he firs place i means elecromagneic waves (e.w.) ravel a he same speed of ligh, secondly i seemed his equaion didn respec he Galileo ransformaions for inerial reference sysems. In fac i was supposed ha he ligh and e.w. propagaed wih he c speed ( km/s) only wih respec o a medium (called eher) which was he absolue reference sysem and wih respec o he oher inerial reference sysems is speed was obained by he heorem of vecor composiion of speeds. The negaive resul of Michelson-Morley s experimen convinced a firs Lorenz o assign srange properies o he eher and o replace Galileo s ransformaions wih differen ransformaions of space-ime and laer Einsein o prove and o accep he same Lorenz ransformaions even hough he claimed he concep of eher wasn necessary. In Special Relaiviy Einsein hen proved he lengh conracion and he ime dilaion, and replaced he heorem of vecor composiion of speeds wih he heorem of addiion of speeds hrough which he proved he speed of boh ligh and elecromagneic waves was always he same wih respec o all he inerial reference sysems. I seemed o be he surmouning wih regard o he conradicion of he fourh equaion. As a maer of fac we proved [1][][3] his isn rue and new conradicions [][3] are caused by boh he Principle of Consancy of he Speed of Ligh and Lorenz s Transformaions ha are he fundamenals of Special Relaiviy.

3 Therefore we wan o consider now he quesion wihin he Theory of Reference Frames [][3][4]. If he equaions (1), (), (3) and (4) are valid wih respec o he reference sysem supposed a res, hen as per he Principle of Relaiviy, which claims physical laws are independen of he considered inerial reference sysem, wih respec o any oher inerial reference sysem he same equaions become div E = (5) div B = 0 (6) ro E = - B (7) ro B = J + 1 E (8) c where = is he inerial ime and c =cöv is he relaivisic speed of ligh. In order o prove he complee equivalence of equaions (5), (6), (7), (8) wih equaions (1), (), (3), (4) we disinguished [][3] beween Galilean reference sysems and Einseinian reference sysems. Galilean reference sysems are closed, isolaed and do no inerac among hem and wih he universe. In ha case, being always wihin he Theory of Reference Frames = E =E, B =B, for he equivalence of he wo groups of equaions [(1),(),(3),(4)] and [(5),(6),(7),(8)], i mus be J = J + E 1 c c Ö v (9) The erm E 1 c c Ö v (10) represens he relaivisic curren densiy and for v<<c i machs [] Ö v E (11) c Einseinian reference sysems are open, no isolaed and ineracive for which in ha case he Lorenz force F L =qu B causes in he moving reference sysem an induced elecric field for cu flux E L =u B (Lorenz s field, is he vecor produc) ha is added o he elecric field E for which E = E + u B (1) 3

4 3. The Lorenz force The exisence of he Lorenz field suggess o make some changes in Maxwell s equaions. We can observe ha in he classic group of equaions he second equaion (divb=0) is indeerminae, and herefore lile revealing, because i is always rue for every value of B for which we can hink o replace i jus wih he Lorenz field and so o obain he following new group of Maxwell s equaions div E = (13) ro B = J + 1 E (14) c ro E = - B (15) E = E + u B (16) In he equaion (16) he Lorenz field is considered a physical even and i represens he force which works on moving elecric charges wih speed u inside a field of magneic inducion B. The equaion of coninuiy of elecric charge div J = - (17) isn independen and can be derived from equaions (13) and (14) because he equaion (13) is valid also for no saic elecric fields when he elecric charge densiy isn consan wih respec o ime. Therefore he group of equaions (13), (14), (15) and (16) is a complee group in order o describe elecromagneism. The Lorenz field can be considered also he cause of elecromagneic inducion due o cu flux wih respec o he S moving reference sysem which has v speed wih respec o he resing reference sysem S (fig.1). z z S S u O O x x v y=y Fig.1 Represenaion of inerial reference sysems. S is a res and S is moving wih v speed. 4

5 As per he Principle of Relaiviy, wih respec o he S inerial reference sysem, he Maxwell equaions are independen of he moion of he reference sysem and herefore he Maxwell equaions become div E = (18) ro B = J + 1 E (19) c ro E = - B (0) E = E + (u+v) B (1) Considering he physical even due only o he cu flux and supposing herefore ha u=0 he equaion (1) becomes E = E + v B () If he elecromagneic wave propagaes along he y axis wih respec o he S resing reference frame, in he Theory of Reference Frames i is From () we deduce = and c = c + v (3) div E = div E + div v B (4) Assuming ha v B=E L (Lorenz s elecric field) we have where L is he Lorenz elecric charge densiy. From (4) we have div E L = L (5) div E = div E + div E L From (0) and () = + L (6) 5

6 ro E = ro E + ro v B = B B + ro v B = B B = B ro v B (7) 0 We deduce from (7) div B = div B = 0. From (19) J = ro B 1 E c J = ro B ro ro v B 1 E + v B 0 (c + v) J = J + 1 E 1 E 1 v B ro ro v B c (c + v) (c + v) 0 J = J + E 1 1 v B 1 ro ro v B (8) 1 + v 1 + v 0 c c For v<<c we have J = J + v E v v B 1 ro ro v B (9) c c 0 The (8) gives he mahemaical expression abou J in order ha Maxwell s equaions respec he Principle of Relaiviy and are invarian wih respec o inerial reference sysems. The (9) gives he same mahemaical expression [] abou J for v<<c. In he Einseinian reference sysems, in addiion o he relaivisic curren densiy, he induced curren densiy mus be considered and i depends on he Lorenz elecric field v B. The (), (3), (6), (7) and (8) represen he equaions of ransformaions for Einseinian inerial reference sysems. 6

7 4. Equaions of he elecromagneic nanofield Maxwell s equaions describe elecromagneic phenomena which are characerized by a propagaion of coninuous elecromagneic waves. Generally i is supposed ha also ligh propagaes by elecromagneic waves bu we proved [5] ligh and he oher radiaions wih higher frequencies (X,, rays) are discree beams made up of energy quana. In he propagaion of boh ligh and radiaions here isn propagaion of a macroscopic elecromagneic field bu only moion of energy quana. We have also proved neverheless ha he single phoon (for he ligh) and in he general he single energy quanum (for oher radiaions) are elecromagneic nanowaves produced by an elecromagneic nanofield ha can be described by Maxwell s equivalen equaions [1] (phoon equaions). The difference beween waves and nanowaves is defined only by he wavelengh: elecromagneic waves include long, medium, shor waves and microwaves. Nanowaves include infrared rays, ligh phoons, ulraviole rays, X rays, rays and rays. Microwaves uilize he wavelengh band (100mm 0,1mm), infrared radiaion uilizes he wavelengh band (1mm 0,8 m). In he lile band (1mm 0,1mm) here is he full ransiion from elecromagneic waves o nanowaves, from microwaves o infrared radiaion, from he coninuous macroscopic srucure of he elecromagneic energy o he disconinuous microscopic srucure of radiaions. Elecromagneic nanowaves are energy quana which ravel wih he velociy of ligh. Equaions of he elecromagneic nanofield conneced wih he single energy quanum can be derived from Maxwell s equaions considering ha he single energy quanum generally is produced by an acceleraed elecron [6] which for example sars from he O origin a ime =0 and moves wih a z consan acceleraion and velociy u z =a z along he z axis (fig.). We can wrie Maxwell s (14) and (16) equaions like his (le us make use of small leers for he elecromagneic nanofield) ro b = u z e + j (30) c z e = e o + u b (31) where j and e o ( e o = j, is he elecric resisiviy) are he curren nanodensiy and he elecric nanofield conneced wih he moving elecron. The equaions (30) and (31) describe he beginning of he elecromagneic nanofield. I is well-known in ha case u b=0, because he u and b vecors are perpendicular, and e=e o. The following equaions (3) and (33) describe insead he propagaion of he elecromagneic nanowave conneced wih he single energy quanum wih respec o he S resing reference sysem 7

8 ro e = - b (3). ro b = 1 e + j (33) c z z S r[x, y, z] S v O O y =y-v y x x y=y Fig. The acceleraed elecron, which generaes he single energy quanum, moves wih u z =a z speed along he z axis. A iniial ime = =0 he wo reference sysems S and S coincide. The single energy quanum moves along he y axis like he S reference frame. Wih respec o he resing reference sysem S we have in he general r(x, y, z)=xi+yj+zk (34) Because for he sake of argumen he elecron moion happens along he z axis and supposing ha he elecron has a poin srucure, which doesn invalidae he correcness of he reasoning, we have x=y=0 and r=zk (35) u(u x, u y, u z ) = r = x i + y j + z k u = u z = dz k (36) d a z = du z d (37) z = a z (38) where dz/d=u z is he speed of he elecron wih respec o S. Wih respec o he S moving inerial reference sysem, he equaions ha describe he beginning of he elecromagneic nanofield become 8

9 ro b = u z e + j (39) c z e = e v b (40) and he equaions ha describe he propagaion become Because we have ro e = b (41) ro b = 1 e + j (4) c r (x, y, z ) = x i + y j + z k x = 0 r = y j + z k (43) u (u x, u y, u z ) = r = x i + y j + z k u = y j + z k (44) where r / =u is he speed of he elecron wih respec o S. According o resuls obained in he Theory of Reference Frames [][4][7] and o resuls here obained relaive o he elecromagneic nanofield, he equaions of ransformaions from S o S in order o describe boh he beginning and he propagaion of he elecromagneic nanofield are = (inerial ime) c = c v (relaivisic speed of ligh) x = x = 0 ; y = y v ; z = z = a z ; u x = u x = 0 ; u y = u y v ; u z = u z = a z ; 9

10 e = e v b = e o + (u v) b b = b + ro v b 0 j = j + e 1 1 v b + 1 ro ro v b (45) 1 v 1 v 0 c c For v<<c he (44) equaion becomes j = j v e + 1 v v b + 1 ro ro v b (46) c c 0 The heory of he elecromagneic nanofield, ha we have here developed, is fine wheher for aomic energy quana (infrared rays, ligh phoons, ulraviole rays, X rays) or for energy quana due o paricle ransformaions ( rays and rays ). Consequenly his heory is valid also for neurinos ha are elecromagneic nanowaves [7][8] belonging o he gamma and dela radiaion. We have also seen [4][7] ha moving elecrodynamic paricles have a ime relaivisic effec because of he variaion of elecrodynamic mass wih he speed bu his effec concerns only he average lifes of paricles wih respec o he resing reference frame and no he kinemaic inerial ime ha is he same for all inerial reference frames. References [1] D. Sasso, On he physical srucure of radian energy: waves and corpuscles, vixra.org, id: , 010 [] D. Sasso, Relaivisic Effecs of he Theory of Reference Frames, Physics Essays, Volume 0 (007), No.1 [3] D. Sasso, Is he Speed of Ligh Invarian or Covarian?, arxiv.org, id: , 010 [4] D. Sasso, Dynamics and Elecrodynamics of Moving Real Sysems in he Theory of Reference Frames, arxiv.org, id: , 010 [5] D. Sasso, Phoon Diffracion, vixra.org, id: , 010 [6] D. Sasso, On Primary Physical Transformaions of Elemenary Paricles: he Origin of Elecric Charge, vixra.org, id: , 01 [7] D. Sasso, If he Speed of Ligh is No an Universal Consan According o he Theory of Reference Frames: on he Physical Naure of Neurino, vixra.org: , 011 [8] D. Sasso; Bea Radiaion, Gamma Radiaion and Elecron Neurino in he Process of Neuron Decay; vixra.org, id: , 01 10